I have turned my monitor physically in portrait position. I adjusted Settings -> Displays so that the layout for my desktop is okay, but the login screen is still landscape.

I read that this problem could be fixed with following instructions:

sudo cp ~/.config/monitors.xml /var/lib/gdm3/.config/
sudo chown gdm:gdm /var/lib/gdm3/.config/monitors.xml
sudo systemctl restart gdm3

But to no avail, it just does not work. No matter if I reboot or even completely turn off and back on computer. The login screen remains in landscape position.

According to mutter documentation monitors.xml is read from /etc/xdg. Copied file there but it did not work either. This is clearly an issue that cannot be fixed with blind workarounds.

I edited /etc/xdg/monitors.xml so that there were different settings than in ~/.config/monitors.xml. After that some settings were applied in gdm process login screen while the rotation - it just wont budge, that change havae effects only in user process desktop. The problem is not the location of monitors.xml file, but how it takes effects on different processess, gdm land or user space.

Is it so that mutter under gdm does not have same capabilities than under user process desktop? If so, this is just banging head on the wall.

Could someone please explain the architecture behind all this gdm/mutter/gnome-shell mess?

I have not tested this solution. It is based on this GDM Archwiki.

The original poster said this solution did not work for him.

Try the following

First, let's removed the file you have copied from the copied location:

sudo rm /var/lib/gdm3/.config/monitors.xml

Removing this file from this location shouldn't cause any problem, as it does not do anything.

Second, Let's open a TTY:

To open a TTL, hold down Ctrl+Alt and press the F3 key. If you are using a laptop's keyboard you may have to hold down Ctrl+Alt+Fn together.

Login to the TTY using your userID and password.

Third, stop gdm3, and copy the file to the new location.

sudo systemctl stop gdm3
sudo cp ~/.config/monitors.xml /etc/xdg/monitors.xml
sudo chown root:root /etc/xdg/monitors.xml
sudo systemctl start gdm3
exit

You may return to the GDM login screen by holding down Ctrl+Alt and press the F1 key.

If restarting gdm3 does not work, try rebooting the computer.

Let me know in comments below if this works or not and if you had to reboot.

Let's hope this will help